Stephane Gaubert (INRIA)
Title: Tropical modules, zero-sum games and non-archimedean
optimization
Abstract: Modules over the Boolean or tropical semifield have appeared in
recent work of Connes and Consani, developing homological algebra in
characteristic one. In this talk, I will highlight the role of such modules in
the study of open issues in computational complexity. One first question
concerns the polynomial time solvability of zero-sum games with ergodic
payments (mean payoff games). Another question, known as Smale
Problem 9, concerns the complexity of linear programming over the real
numbers. Tropical geometry allows one to relate both problems: mean
payoff games turn out to be equivalent to “generic” linear programs over
nonarchimedean fields. In this way, we transfer complexity results from
linear programming to mean payoff games. Moreover, we obtain a counter
example showing that one of the main linear programming methods
(interior points) can make an exponential number of arithmetic
operations. This is based on a series of works with Akian, Allamigeon,
Benchimol and Joswig.